Table Of ContentLicensed to:
Licensed to:
Business Mathematics
and Statistics
Andre Francis BSc MSc
Perinatal Institute
Birmingham
Andre Francis works as a medical statistician.
He has previously taught Mathematics, Statistics
and Information Processing to students on busi-
ness and professional courses. His teaching experi-
ence has covered a wide area, including training
students learning basic skills through to teaching
undergraduates. He has also had previous indus-
trial (costing) and commercial (export) experience
and served for six years in statistical branches of
Training Command in the Royal Air Force.
Sixth Edition
Australia • Canada • Mexico • Singapore • Spain • United Kingdom • United States
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Acknowledgements
The author would like to express thanks to the many students and teachers who have
contributed to the text in various ways over the years.
In particular he would like to thank the following examining bodies for giving permis-
sion to reproduce selected past examination questions:
Chartered Association of Certified Accountants (ACCA)
Chartered Institute of Management Accountants (CIMA)
Institute of Chartered Secretaries and Administrators (ICSA)
Chartered Institute of Insurance (CII)
Association of Accounting Technicians (AAT)
Each question used is cross referenced to the appropriate Institute or Association.
A CIP catalogue record for this book is available from the British Library
First Edition 1986
Second Edition 1988; Reprinted 1990; Reprinted 1991
Third Edition 1993; Reprinted 1993
Fourth Edition 1995; Reprinted 1996; Reprinted 1997
Fifth Edition 1998; Reprinted 2003 by Thomson Learning
Sixth Edition 2004; Published by Thomson Learning
Copyright A. Francis © 2004
ISBN 1-84480-128-4
All rights reserved
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by
any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the
copyright owner except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or
under the terms of a licence issued by The Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London
W1P 9HE. Applications for the copyright owner’s permission to reproduce any part of this publication should
be addressed to the publisher.
Typeset in Nottingham, UK by Andre Francis
Printed in Croatia by Zrinski d.d.
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Contents
Preface....................................................................................................................................v
1 Introduction to Business Mathematics and Statistics.............................1
Part 1 Data and their presentation...................................................................................5
2 Sampling and Data Collection...................................................................6
3 Data and their Accuracy............................................................................24
4 Frequency Distributions and Charts.......................................................38
5 General Charts and Graphs......................................................................63
Examination questions......................................................................................90
Part 2 Statistical measures...............................................................................................95
6 Arithmetic Mean........................................................................................96
7 Median.......................................................................................................107
8 Mode and Other Measures of Location................................................117
9 Measures of Dispersion and Skewness.................................................129
10 Standard Deviation..................................................................................136
11 Quantiles and the Quartile Deviation...................................................148
Examination example and questions............................................................159
Part 3 Regression and correlation................................................................................165
12 Linear Functions and Graphs.................................................................166
13 Regression Techniques............................................................................173
14 Correlation Techniques...........................................................................191
Examination examples and questions...........................................................207
Part 4 Time series analysis.............................................................................................213
15 Time Series Model....................................................................................214
16 Time Series Trend.....................................................................................219
17 Seasonal Variation and Forecasting.......................................................229
Examination example and questions............................................................242
Part 5 Index numbers......................................................................................................247
18 Index Relatives.........................................................................................248
19 Composite Index Numbers.....................................................................259
20 Special Published Indices........................................................................272
Examination questions....................................................................................281
Part 6 Compounding, discounting and annuities.....................................................285
21 Interest and Depreciation........................................................................286
22 Present Value and Investment Appraisal.............................................302
23 Annuities...................................................................................................318
Examination examples and questions...........................................................330
iii
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Contents
Part 7 Business equations and graphs 337
24 Functions and Graphs 338
25 Linear Equations 351
26 Quadratic and Cubic Equations 364
27 Differentiation and Integration 374
28 Cost, Revenue and Profit Functions 385
Examination examples and questions 395
Part 8 Probability 403
29 Set Theory and Enumeration 404
30 Introduction to Probability 419
31 Conditional Probability and Expectation 436
Examination examples and questions 449
Part 9 Further probability 455
32 Combinations and Permutations 456
33 Binomial and Poisson Distributions 462
34 Normal Distribution 473
Examination example and questions 490
Part 10 Specialised business applications 495
35 Linear Inequalities 496
36 Matrices 508
37 Inventory Control 526
38 Network Planning and Analysis 543
Examination example and questions 555
Answers to student exercises 562
Answers to examination questions 581
Appendices 650
1 Compounding and Discounting Tables 650
2 Random Sampling Numbers 654
3 Exponential Tables. Values of e–m 655
4 Standard Normal Distribution Tables 657
Index 659
iv
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Preface
1. Aims of the book
The general aim of the book is to give a thorough grounding in basic Mathematical
and Statistical techniques to students of Business and Professional studies. No prior
knowledge of the subject area is assumed.
2. Courses covered
a) The book is intended to support the courses of the following professional
bodies:
Chartered Association of Certified Accountants
Chartered Institute of Management Accountants
Institute of Chartered Secretaries and Administrators
b) The courses of the following bodies which will be supported by the book to a
large extent:
Chartered Institute of Insurance
Business and Technical Education Council (National level)
Association of Accounting Technicians
c) The book is also meant to cater for the students of any other courses who
require a practical foundation of Mathematical and Statistical techniques used
in Business, Commerce and Industry.
3. Format of the book
The book has been written in a standardised format as follows:
a) There are TEN separate parts which contain standard examination testing
areas.
b) Numbered chapters split up the parts into smaller, identifiable segments, each
of which have their own Summaries and Points to Note.
c) Numbered sections split the chapters up into smaller logical elements involving
descriptions, definitions, formulae or examples.
At the end of each chapter, there is a Student Self Review section which contains
questions that are meant to test general concepts, and a Student Exercise section
which concentrates on the more practical numerical aspects covered in the chapter.
At the end of each part, there is
a) a separate section containing examination examples with worked solutions
and
b) examination questions from various bodies. Worked solutions to these ques-
tions are given at the end of the book.
v
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Preface
4. How to use the book
Chapters in the book should be studied in the order that they occur.
After studying each section in a chapter, the Summaries and Points to Note should
be checked through. The Student Self Review Questions, which are cross-refer-
enced to appropriate sections, should first be attempted unaided, before checking
the answers with the text. Finally the Student Exercises should be worked through
and the answers obtained checked with those given at the end of the book.
After completing a particular part of the book, the relevant section of the examina-
tion questions (at the end of the book) should be attempted. These questions should
be considered as an integral part of the book, all the subject matter included having
been covered in previous chapters and parts. Always make some attempt at the
questions before reading the solution.
5. The use of calculators
Examining bodies permit electronic calculators to be used in examinations. It is
therefore essential that students equip themselves with a calculator from the begin-
ning of the course.
Essential facilities that the calculator should include are:
a) a square root function, and
b) an accumulating memory.
Very desirable extra facilities are:
c) a power function (labelled ‘xy’),
d) a logarithm function (labelled ‘log x’), and
e) an exponential function (labelled ‘ex’).
Some examining bodies exclude the use (during examinations) of programmable
calculators and/or calculators that provide specific statistical functions such as the
mean or the standard deviation. Students are thus urged to check on this point
before they purchase a calculator. Where relevant, this book includes sections
which describe techniques for using calculators to their best effect.
Andre Francis, 2004
vi
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Answers to examination questions
Part 1
Question 1
Simple random sampling. A method of sampling whereby each member of the popu-
lation has an equal chance of being chosen. Normally, random sampling numbers
are used to select individual items from some defined sampling frame.
Stratification. This is a process which splits a population up into as many groups
and sub-groups (strata) as are of significance to the investigation. It can be used as
a basis for quota sampling, but more often is associated with stratified (random)
sampling. Stratified sampling involves splitting the total sample up into the same
proportions and groups as that for the population stratification and then sepa-
rately taking a simple random sample from each group. For example, employees
of a company could be split into male/female, full-time/part-time and occupation
category.
Quota sampling. A method of non-random sampling which is popular in market
research. It uses street interviewers, armed with quotas of people to interview in
a range of groups, to collect information from passers-by. For example, obtaining
peoples’ attitudes regarding the worth of secondary double glazing.
Sample frame. This is a listing of the members of some target population which
needs to be used in order to select a random sample. An example of a sampling
frame would be a stock list, if a random sample was required from current ware-
house stock.
Cluster sampling. This is another non-random method of sampling, used where no
sampling frame is in evidence. It consists of selecting (randomly) one or more areas,
within which all relevant items or subjects are investigated. For example, a cluster
sample could be taken in a large town to interview tobacconists.
Systematic sampling. A quasi-random method of sampling which involves examining
or interviewing every n-th member of a population. Very useful method where no
sampling frame exists, but population members are physically in evidence and
ordered. For example, items coming off a production line. It is virtually as good as
random sampling except where the items or members repeat themselves at regular
intervals, which could lead to serious bias.
Question 2
(a) A postal questionnaire is a much cheaper and more convenient method of
collecting data than the personal interview and often very large samples can be
taken. However, much more care must be taken in the design of the questions,
since there will be no help to hand if questions seem ambiguous or personal to
the respondent. Also the response rate is very low, sometimes less than 20%,
but this can sometimes be made larger by free gifts or financial incentives.
The personal interview has the particular advantage that difficult or ambiguous
questions can be explained as well as the fact that an interviewer can make
allowances or small adjustments according to the situation. Also, the question-
naire will be filled in as required. Disadvantages of this method include the cost,
581
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Answers to examination questions – part 1
the fact that large samples cannot generally be undertaken and the training of
interviewers.
(b) Simple random sampling has the particular advantage that the method of selec-
tion (normally through the use of random sampling numbers) is free from bias.
That is, each member of the population has an equal chance of being chosen
as part of the sample. However, it cannot be guaranteed that the sample itself
is truly representative of the population. For example, if a human population
being sampled comprised 48% males, it is unlikely that the sample would
reflect this percentage exactly.
Quota sampling is not a random sampling method and thus is generally at a
disadvantage with regard to obtaining information that can claim to be repre-
sentative. However, if the population has been stratified reasonably well, the
street interviewer is experienced and conscientious and the questioning sites
have been well thought out, it could be argued that, in certain localised situ-
ations, a quota sample could be very representative. For example, to gauge
peoples opinions of a new shopping centre or to find out the views of theatre-
goers about a particular theatre.
Question 3
(a) (i) See pie chart.
Real consumers' relative expenditure in
1984 - component categories (1980 prices)
Durable
goods
Other
services
Food
Rent and Alcohol and
rates tobacco
Clothing and
Other goods footwear
Energy
products
(ii) Other goods: books, toys, toiletries, transport. Other services: insurance,
recreation, entertainment, (private) dental/health care.
582
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Licensed to:
Answers to examination questions – part 1
(b) See line diagram.
Comparative profit before tax (1980=100)
of six Scotch whisky companies
Bells
200
180 Macallan
MacDonald
160
Highland
140
120
Distillers
100
Invergordon
80
1979 1980 1981 1982 1983 1984 1985
Year
Question 4
(a) i. An absolute error is the difference between an estimated value and its
true value. In most cases, only a maximum absolute error will be able to
be calculated. For example, if a company’s yearly profit was quoted as
£252,000 (to the nearest £1000), the maximum absolute error would be
£500.
ii. A relative error is an absolute error expressed as a percentage of the given
estimated value. Thus in the example above, the maximum relative error in
the company’s yearly profit is:
500
×100% = 0.2%
252,000
iii. A compensating error is an error that is made when ‘fair’rounding has
been carried out. For example, the numbers of people employed in each of
a number of factories might well be rounded fairly, to the nearest 1000, say.
When numbers, subject to compensating errors, are added, the total relative
error should be approximately zero.
iv. Biased errors are made if rounding is always carried out in one direction.
For example, when people’s ages are quoted, they are normally rounded
down to the lowest year. The error in the sum of numbers that are subject to
biased errors is relatively high.
583
Copyright 2004 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
